164 XIII INTEGRATION 2. Let A be a subset of I = [0, 1] which is not Lebesgue-measurable, and let / be the function which is equal to 1 on A and to 1 on I A. Then f2 is integrable but / is not measurable. 3. Consider the sequence (An) denned in Problem 4(b) of Section 13.8, and let H be the union of the sets A2n+i (n > 0). Show that, if J is any nonempty open interval con- tained in I, then the sets J n H and J n (I H) both have strictly positive measure. Now let / be a function equal to 9?H almost everywhere. Show that there does not exist any sequence (/) of continuous functions on I which converges at every point of I to the function/. (Remark that /must be discontinuous at every point of I, and use Problem 3 of Section 12.16.) 4. Let ju, be a bounded measure of total mass 1 on a locally compact space X. Let (A/Oiscfc^n be a finite sequence of integrable subsets of X, such that /z(Ak) c> 0 for all k. If nc> 1 and e > c(l c)/(n 1), show that there exist two distinct indices ij such that ft(Aj n Aj) ^ c2 e. (Argue by contradiction, using the Cauchy- Schwarz inequality.) 5. Let (/) be a sequence of functions belonging to ^a(X, JJL) (p = 1 or 2) which converges almost everywhere to a function /. (a) Show that, if Np(/w) <; a for all n, then /e tf&Qi, ju) and Np(/) ^ a. (Use Fatou's lemma.) Give an example (withp = 1) where Np(/n) does not tend to Np(/). (b) Now suppose in addition that Np(/n) -> Np(/) as w-> + oo. Show that Np(/ /) -> 0. (Show that for each e > 0 there exists a compact subset K of X and an integer n0 such that \fn \ p dp, ^ e for all n ^ Ğ0 .) 6. In the space ^R(!,A), where I = ]0, 1] and A is Lebesgue measure, consider the functions /* (a real), which belong to this space provided that a > £. Show that a sequence of distinct exponents an > i is such that the functions fn form a total sequence in J2PR(I, A) if and only if the sequence (an) satisfies one of the following three conditions: (1) there exists a subsequence of the sequence (an) tending to a finite limit > i; (2) lim an = - and n-voo 2 (3) lim an = + oo ; -fco, n-*oo (Using Weierstrass' theorem (7.4.1 ), calculate for each integer m > 0 the minimum of N2(^ /(/)) as /runs through the set of all linear combinations of the first n functions t"k. For this purpose, use Problem 3(b) of Section 6.6, and the formula (Cauchy's determinant) det El (Ği + h) i,J in which (af), (bi) are any two sequences of n distinct numbers >0.) Deduce that then the normed space L£ .Use (a) to prove that the condition is necessary.)